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## G = C4⋊(D6⋊C4)  order 192 = 26·3

### The semidirect product of C4 and D6⋊C4 acting via D6⋊C4/C22×S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C4⋊(D6⋊C4)
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — S3×C23 — S3×C22×C4 — C4⋊(D6⋊C4)
 Lower central C3 — C2×C6 — C4⋊(D6⋊C4)
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for C4⋊(D6⋊C4)
G = < a,b,c,d | a4=b6=c2=d4=1, ab=ba, ac=ca, dad-1=a-1, cbc=b-1, bd=db, dcd-1=b3c >

Subgroups: 632 in 234 conjugacy classes, 87 normal (41 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C23, C23, Dic3, C12, C12, D6, D6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C23×C4, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, S3×C2×C4, S3×C2×C4, C22×Dic3, C22×Dic3, C22×C12, C22×C12, S3×C23, C23.7Q8, C6.C42, C2×C4⋊Dic3, C2×D6⋊C4, C6×C4⋊C4, S3×C22×C4, C4⋊(D6⋊C4)
Quotients:

Smallest permutation representation of C4⋊(D6⋊C4)
On 96 points
Generators in S96
(1 76 16 72)(2 77 17 67)(3 78 18 68)(4 73 13 69)(5 74 14 70)(6 75 15 71)(7 39 95 35)(8 40 96 36)(9 41 91 31)(10 42 92 32)(11 37 93 33)(12 38 94 34)(19 65 29 55)(20 66 30 56)(21 61 25 57)(22 62 26 58)(23 63 27 59)(24 64 28 60)(43 89 53 79)(44 90 54 80)(45 85 49 81)(46 86 50 82)(47 87 51 83)(48 88 52 84)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)(25 26 27 28 29 30)(31 32 33 34 35 36)(37 38 39 40 41 42)(43 44 45 46 47 48)(49 50 51 52 53 54)(55 56 57 58 59 60)(61 62 63 64 65 66)(67 68 69 70 71 72)(73 74 75 76 77 78)(79 80 81 82 83 84)(85 86 87 88 89 90)(91 92 93 94 95 96)
(1 15)(2 14)(3 13)(4 18)(5 17)(6 16)(7 93)(8 92)(9 91)(10 96)(11 95)(12 94)(19 26)(20 25)(21 30)(22 29)(23 28)(24 27)(31 41)(32 40)(33 39)(34 38)(35 37)(36 42)(43 53)(44 52)(45 51)(46 50)(47 49)(48 54)(55 62)(56 61)(57 66)(58 65)(59 64)(60 63)(67 74)(68 73)(69 78)(70 77)(71 76)(72 75)(79 89)(80 88)(81 87)(82 86)(83 85)(84 90)
(1 48 24 36)(2 43 19 31)(3 44 20 32)(4 45 21 33)(5 46 22 34)(6 47 23 35)(7 71 87 59)(8 72 88 60)(9 67 89 55)(10 68 90 56)(11 69 85 57)(12 70 86 58)(13 49 25 37)(14 50 26 38)(15 51 27 39)(16 52 28 40)(17 53 29 41)(18 54 30 42)(61 93 73 81)(62 94 74 82)(63 95 75 83)(64 96 76 84)(65 91 77 79)(66 92 78 80)

G:=sub<Sym(96)| (1,76,16,72)(2,77,17,67)(3,78,18,68)(4,73,13,69)(5,74,14,70)(6,75,15,71)(7,39,95,35)(8,40,96,36)(9,41,91,31)(10,42,92,32)(11,37,93,33)(12,38,94,34)(19,65,29,55)(20,66,30,56)(21,61,25,57)(22,62,26,58)(23,63,27,59)(24,64,28,60)(43,89,53,79)(44,90,54,80)(45,85,49,81)(46,86,50,82)(47,87,51,83)(48,88,52,84), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,15)(2,14)(3,13)(4,18)(5,17)(6,16)(7,93)(8,92)(9,91)(10,96)(11,95)(12,94)(19,26)(20,25)(21,30)(22,29)(23,28)(24,27)(31,41)(32,40)(33,39)(34,38)(35,37)(36,42)(43,53)(44,52)(45,51)(46,50)(47,49)(48,54)(55,62)(56,61)(57,66)(58,65)(59,64)(60,63)(67,74)(68,73)(69,78)(70,77)(71,76)(72,75)(79,89)(80,88)(81,87)(82,86)(83,85)(84,90), (1,48,24,36)(2,43,19,31)(3,44,20,32)(4,45,21,33)(5,46,22,34)(6,47,23,35)(7,71,87,59)(8,72,88,60)(9,67,89,55)(10,68,90,56)(11,69,85,57)(12,70,86,58)(13,49,25,37)(14,50,26,38)(15,51,27,39)(16,52,28,40)(17,53,29,41)(18,54,30,42)(61,93,73,81)(62,94,74,82)(63,95,75,83)(64,96,76,84)(65,91,77,79)(66,92,78,80)>;

G:=Group( (1,76,16,72)(2,77,17,67)(3,78,18,68)(4,73,13,69)(5,74,14,70)(6,75,15,71)(7,39,95,35)(8,40,96,36)(9,41,91,31)(10,42,92,32)(11,37,93,33)(12,38,94,34)(19,65,29,55)(20,66,30,56)(21,61,25,57)(22,62,26,58)(23,63,27,59)(24,64,28,60)(43,89,53,79)(44,90,54,80)(45,85,49,81)(46,86,50,82)(47,87,51,83)(48,88,52,84), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,15)(2,14)(3,13)(4,18)(5,17)(6,16)(7,93)(8,92)(9,91)(10,96)(11,95)(12,94)(19,26)(20,25)(21,30)(22,29)(23,28)(24,27)(31,41)(32,40)(33,39)(34,38)(35,37)(36,42)(43,53)(44,52)(45,51)(46,50)(47,49)(48,54)(55,62)(56,61)(57,66)(58,65)(59,64)(60,63)(67,74)(68,73)(69,78)(70,77)(71,76)(72,75)(79,89)(80,88)(81,87)(82,86)(83,85)(84,90), (1,48,24,36)(2,43,19,31)(3,44,20,32)(4,45,21,33)(5,46,22,34)(6,47,23,35)(7,71,87,59)(8,72,88,60)(9,67,89,55)(10,68,90,56)(11,69,85,57)(12,70,86,58)(13,49,25,37)(14,50,26,38)(15,51,27,39)(16,52,28,40)(17,53,29,41)(18,54,30,42)(61,93,73,81)(62,94,74,82)(63,95,75,83)(64,96,76,84)(65,91,77,79)(66,92,78,80) );

G=PermutationGroup([[(1,76,16,72),(2,77,17,67),(3,78,18,68),(4,73,13,69),(5,74,14,70),(6,75,15,71),(7,39,95,35),(8,40,96,36),(9,41,91,31),(10,42,92,32),(11,37,93,33),(12,38,94,34),(19,65,29,55),(20,66,30,56),(21,61,25,57),(22,62,26,58),(23,63,27,59),(24,64,28,60),(43,89,53,79),(44,90,54,80),(45,85,49,81),(46,86,50,82),(47,87,51,83),(48,88,52,84)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24),(25,26,27,28,29,30),(31,32,33,34,35,36),(37,38,39,40,41,42),(43,44,45,46,47,48),(49,50,51,52,53,54),(55,56,57,58,59,60),(61,62,63,64,65,66),(67,68,69,70,71,72),(73,74,75,76,77,78),(79,80,81,82,83,84),(85,86,87,88,89,90),(91,92,93,94,95,96)], [(1,15),(2,14),(3,13),(4,18),(5,17),(6,16),(7,93),(8,92),(9,91),(10,96),(11,95),(12,94),(19,26),(20,25),(21,30),(22,29),(23,28),(24,27),(31,41),(32,40),(33,39),(34,38),(35,37),(36,42),(43,53),(44,52),(45,51),(46,50),(47,49),(48,54),(55,62),(56,61),(57,66),(58,65),(59,64),(60,63),(67,74),(68,73),(69,78),(70,77),(71,76),(72,75),(79,89),(80,88),(81,87),(82,86),(83,85),(84,90)], [(1,48,24,36),(2,43,19,31),(3,44,20,32),(4,45,21,33),(5,46,22,34),(6,47,23,35),(7,71,87,59),(8,72,88,60),(9,67,89,55),(10,68,90,56),(11,69,85,57),(12,70,86,58),(13,49,25,37),(14,50,26,38),(15,51,27,39),(16,52,28,40),(17,53,29,41),(18,54,30,42),(61,93,73,81),(62,94,74,82),(63,95,75,83),(64,96,76,84),(65,91,77,79),(66,92,78,80)]])

48 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 6A ··· 6G 12A ··· 12L order 1 2 ··· 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 6 6 6 6 2 2 2 2 2 4 4 4 4 6 6 6 6 12 12 12 12 2 ··· 2 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + - + + + - - + image C1 C2 C2 C2 C2 C2 C4 S3 D4 D4 Q8 D6 C4○D4 C4×S3 D12 C3⋊D4 S3×D4 D4⋊2S3 S3×Q8 Q8⋊3S3 kernel C4⋊(D6⋊C4) C6.C42 C2×C4⋊Dic3 C2×D6⋊C4 C6×C4⋊C4 S3×C22×C4 S3×C2×C4 C2×C4⋊C4 C2×C12 C22×S3 C22×S3 C22×C4 C2×C6 C2×C4 C2×C4 C2×C4 C22 C22 C22 C22 # reps 1 2 1 2 1 1 8 1 4 2 2 3 4 4 4 4 1 1 1 1

Matrix representation of C4⋊(D6⋊C4) in GL6(𝔽13)

 4 11 0 0 0 0 2 9 0 0 0 0 0 0 5 0 0 0 0 0 0 8 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 1 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 12 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 12 0 0 0 0 0 0 0 11 9 0 0 0 0 4 2

G:=sub<GL(6,GF(13))| [4,2,0,0,0,0,11,9,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,11,4,0,0,0,0,9,2] >;

C4⋊(D6⋊C4) in GAP, Magma, Sage, TeX

C_4\rtimes (D_6\rtimes C_4)
% in TeX

G:=Group("C4:(D6:C4)");
// GroupNames label

G:=SmallGroup(192,546);
// by ID

G=gap.SmallGroup(192,546);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,422,387,58,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^6=c^2=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^3*c>;
// generators/relations

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